Chapter 2, Problem 3E

Explanation of Solution

a)

Converting 137 base 10 to base 3 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 3 which may result in lower number than 137. The number that can be subtracted from the given number 137 with the power of 3 is 81. The number 81 is less than 137. So subtract 81 from 137.

$\begin{array}{l}\text{137}\\ {\text{-81=3}}^{\text{4}}\text{x1}\\ \text{------------}\\ \text{56}\end{array}$

Step 2:

The number that can be subtracted from the given number 56 with the power of 3 is 27 and it should be multiplied by 2 in order to get the nearest number of 56. The number 54 is less than 56. So subtract 56 from 54.

$\begin{array}{l}56\\ {\text{-54=3}}^3\text{x2}\\ \text{------------}\\ 2\end{array}$

Step 3:

Take 3 to the power of 2 that is 9. The number 9 is greater than 2. So make the value as 0.

$\begin{array}{l}2\\ {\text{-0=3}}^2\text{x0}\\ \text{------------}\\ 2\end{array}$

Step 4:

Take 3 to power of 1 that is 3

Explanation of Solution

b)

Converting 248 base 10 to base 5 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 5 which may result in lower number than 248. The number that can be subtracted from the given number 248 with the power of 5 is 125. The number 125 is less than 248. So subtract 125 from 248.

$\begin{array}{l}248\\ {\text{-125=5}}^3\text{x1}\\ \text{------------}\\ 123\end{array}$

Step 2:

The number that can be subtracted from the given number 123 with the power of 5 is 25 and it should be multiplied by 4 in order to get the nearest number of 123. The number 100 is less than 123. So subtract 100 from 123.

$\begin{array}{l}123\\ {\text{-100=5}}^2\text{x4}\\ \text{------------}\\ 23\end{array}$

Step 3:

The number that can be subtracted from the given number 23 with the power of 5 is 5 and it should be multiplied by 4 in order to get the nearest number of 23

Explanation of Solution

c)

Converting 387 base 10 to base 7 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 7 which may result in lower number than 387. The number that can be subtracted from the given number 387 with the power of 7 is 343. The number 343 is less than 387. So subtract 343 from 387.

$\begin{array}{l}387\\ {\text{-343=7}}^3\text{x1}\\ \text{------------}\\ 44\end{array}$

Step 2:

Take 7 to power of 2 that is 49. The number 49 is greater than 44. So make the value as 0.

$\begin{array}{l}44\\ {\text{-0=7}}^2\text{x0}\\ \text{------------}\\ 44\end{array}$

Step 3:

The number that can be subtracted from the given number 44 with the power of 7 is 7 and it should be multiplied by 6 in order to get the nearest number of 44. The number 42 is less than 44. So subtract 42 from 44

Explanation of Solution

d)

Converting 633 base 10 to base 9 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 9 which may result in lower number than 633. The number that can be subtracted from the given number 633 with the power of 9 is 81 and it should be multiplied by 7 in order to get the nearest number of 633. The number 567 is less than 633. So subtract 567 from 633.

$\begin{array}{l}633\\ {\text{-567=9}}^2\text{x7}\\ \text{------------}\\ 66\end{array}$

Step 2:

The number that can be subtracted from the given number 66 with the power of 9 is 9 and it should be multiplied by 7 in order to get the nearest number of 66. The number 63 is less than 66. So subtract 63 from 66